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Decimal expansion of lemniscate constant B.
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%I #74 Nov 04 2024 01:49:43

%S 5,9,9,0,7,0,1,1,7,3,6,7,7,9,6,1,0,3,7,1,9,9,6,1,2,4,6,1,4,0,1,6,1,9,

%T 3,9,1,1,3,6,0,6,3,3,1,6,0,7,8,2,5,7,7,9,1,3,1,8,3,7,4,7,6,4,7,3,2,0,

%U 2,6,0,7,0,7,1,9,5,7,8,3,5,4,1,7,9,4,2,7,7,8,2,4,4,8,9,6,6,9,4,6,8,7,9,5,3,6

%N Decimal expansion of lemniscate constant B.

%C Also decimal expansion of AGM(1,i)/(1+i).

%C See A085565 for the lemniscate constant A. - _Peter Bala_, Oct 25 2019

%C Also the ratio of height to diameter of a "Mylar balloon" (see Paulsen). - _Jeremy Tan_, May 05 2021

%D J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1998.

%H W. H. Paulsen, <a href="https://doi.org/10.2307/2975161">What Is the Shape of a Mylar Balloon?</a>, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.

%H J. Todd, <a href="https://doi.org/10.1145/360569.360580">The lemniscate constants</a>, Comm. ACM, Vol. 18, No. 1 (1975), pp. 14-19; corrigendum, Vol. 18, No. 8 (1975), p. 462.

%H J. Todd, <a href="https://doi.org/10.1007/978-1-4757-3240-5_45">The lemniscate constants</a>, in Pi: A Source Book, pp. 412-417.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-Geometric Mean</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mylar_balloon_(geometry)">Mylar balloon</a>

%H Wolfram Research, <a href="http://functions.wolfram.com/EllipticFunctions/ArithmeticGeometricMean/">Arithmetic-Geometric Mean</a>.

%F Equals (2*Pi)^(-1/2)*GAMMA(3/4)^2.

%F Equals ee/sqrt(2)-1/2*sqrt(2*ee^2-Pi) where ee = EllipticE(1/2), or also Product_{m>=1} ((2*m)/(2*m-1))^(-1)^m. - _Jean-François Alcover_, Sep 02 2014, after Steven Finch.

%F Equals sqrt(2) * Pi^(3/2) / GAMMA(1/4)^2. - _Vaclav Kotesovec_, Oct 03 2019

%F From _Peter Bala_, Oct 25 2019: (Start)

%F Equals 1 - 1/3 - 1/(3*7) - (1*3)/(3*7*11) - (1*3*5)/(3*7*11*15) - ... = hypergeom([-1/2,1],[3/4],1/2) by Gauss’s second summation theorem.

%F Equivalently, define a sequence of rational numbers r(n) recursively by r(n) = (2*n - 3)/(4*n - 1)*r(n-1) with r(0) = 1. Then the constant equals Sum_{n >= 0} r(n) = 1 - 1/3 - 1/21 - 1/77 - 1/231 - 1/627 - 3/4807 - 1/3933 - 13/121923 - 13/284487 - 17/853461 - .... The partial sum of the series to 100 terms gives the constant correct to 32 decimal places.

%F Equals (1/3) + (1*3)/(3*7) + (1*3*5)/(3*7*11) + ... = (1/3) * hypergeom ([3/2,1],[7/4],1/2). (End)

%F Equals (1/2) * A053004. - _Amiram Eldar_, Aug 26 2020

%F Equals (2/3) * 1/A243340. - _Peter Bala_, Mar 25 2024

%F Equals Product_{n>=1} exp(((-1)^n*beta(n))/n), where beta(n) is the Dirichlet beta function. - _Antonio Graciá Llorente_, Oct 16 2024

%e 0.599070117367796103719961246140161939113606331607825779131837476473202607...

%e AGM(1,i) = 0.59907011736779610371... + 0.59907011736779610371...*i

%t RealDigits[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 111]]] [[1]]

%t RealDigits[N[Pi/(4 EllipticK[-1]), 106]][[1]] (* _Jean-François Alcover_, Jun 02 2019 *)

%o (PARI) real(agm(1,I)/(1+I)) \\ _Charles R Greathouse IV_, Mar 03 2016

%o (PARI) (2*Pi)^(-1/2)*gamma(3/4)^2 \\ _Michel Marcus_, Nov 10 2017

%Y Cf. A053004, A076391, A076392, A085565, A243340.

%K nonn,cons,changed

%O 0,1

%A _Robert G. Wilson v_, Oct 09 2002

%E Edited by _N. J. A. Sloane_, Nov 01 2008 at the suggestion of _R. J. Mathar_